... mesh^2.1

The term ``background mesh'' will be explained in Section 3.3.

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... method^2.2

See Section 4.1.2.

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... points^3.1

See Chapter 5 for a description of the Voronoi diagram and the Delaunay properties.

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... M-matrix^3.2

A real, nonsingular $n \times n$ matrix $A$ where $a_{i,j} \leq 0 \ \ \forall \ \ i \neq j$ and $A^{-1} > 0$.

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... space^3.3

A slightly different concept of control space is given in [55].

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... representation^3.4

An image which consists of pixels in two and voxels in three dimensions.

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... point^4.1

Quadrilateral/hexahedral elements require more points.

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... mesh^4.2

For instance see Section 5.6.

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... algorithms^4.3

Section 5.3.

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... topology^4.4

This is strictly only valid for a unique Delaunay Triangulation, Section 5.4 and 6.4.3.

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... points^4.5

The constrained Delaunay Triangulation can only be guaranteed in two dimensions, see Section 5.5.

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... quality^4.6

The max-min and min-max-min Delaunay properties are discussed in Section 5.1.

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... four^5.1

If in three dimensions more than four Voronoi boxes are incident at a Voronoi point, the Delaunay Triangulation is non-unique.

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... NP-complete^5.2

Many outstanding and famous problems exist in computer science for which no deterministic polynomial time algorithms are known. Exponential time algorithms are useless regardless of the speed of a computer, because finding a solution is in any case too expensive. Problems which can be solved by deterministic algorithms in polynomial time are said to be in P. Problems which can only be solved by nondeterministic algorithms in polynomial time are said to be in NP. In other words if the solution is not efficiently found but guessed and then checked for validity by a polynomial time algorithm the problem is in NP. The trouble is that no one has been able to prove that a problem is in NP and not in P. It is unclear whether an efficient deterministic polynomial time algorithm for a problem in NP remains undiscovered or simply does not exist. Problems exist which can be proven to represent all problems in NP. These problems are said to be NP-complete. If an NP-complete problem could be solved by a deterministic polynomial time algorithm, it would be proven that P=NP. This would mean that to all outstanding and famous problems efficient but undiscovered solutions exist. Otherwise one must rely on heuristics and hope that they do not fail for most practical instances of the problem. Some well known NP-complete problems are traveling-salesman, Hamilton-cycle, satisfiability, and longest-path.

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